Large Deviations for Expanding Transformations with ... - UFRGS

0 downloads 0 Views 161KB Size Report
1311. 0022-4715В00В0300-131118.00В0. 2000 Plenum Publishing Corporation. 1 Instituto de Matematica, UFRGS, Porto Alegre, RS, CEP 91500-000, Brazil.
Journal of Statistical Physics, Vol. 98, Nos. 56, 2000

Large Deviations for Expanding Transformations with Additive White Noise S. C. Carmona 1 and A. Lopes 1 Received May 17, 1999; final October 14, 1999 Large-deviations estimates for the autocorrelations of order k of the random process Z n =,(X n )+! n , n0, are obtained. The processes (X n ) n0 and (! n ) n0 are independent, ! n , n0, are i.i.d. bounded random variables, X n =T n(X 0 ), n # N, T: M  M is expanding leaving invariant a Gibbs measure on a compact set M, and ,: M  R is a continuous function. A possible application of this result is the case where M is the unit circle and the Gibbs measure is the one absolutely continuous with respect to the Lebesgue measure on the circle. The case when T is a uniquely ergodic map was studied in Carmona et al. (1998). In the present paper T is an expanding map. However, it is possible to derive large-deviations properties for the autocorrelations samples (1n)  n&1 j=0 Z j Z j+k . But the deviation function is quite different from the uniquely ergodic case because it is necessary to take into account the entropy of invariant measures for T as an important information. The method employed here is a combination of the variational principle of the thermodynamic formalism with Donsker and Varadhan's large-deviations approach. KEY WORDS: Level-2 large deviations; expanding maps; Gibbs states; entropy; Markov process; additive white noise.

1. INTRODUCTION Suppose that (V n ) n0 is a random process in the probability space (0, F, P) where each V n takes values in a locally compact metric space S (the phase space of the process). Let M1(S) be the space of probability measures on B(S), endowed with the weak topology; the set B(S) is the _-field of the Borel subsets of S.

1

Instituto de Matematica, UFRGS, Porto Alegre, RS, CEP 91500-000, Brazil. 1311 0022-4715000300-131118.000  2000 Plenum Publishing Corporation

1312

Carmona and Lopes

The empirical means for (V n ) n0 are L n(w, } )=

1 n&1 : $ V (w)( } ), n k=0 k

w#0

(1.1)

The level-2 large deviation theory for (V n ) n0 deals with estimates of the type lim

1 P(L n # F ) & inf I(&) n &#F

(1.2)

1 ln P(L n # G) & inf I(&) n &#G

(1.3)

n  +

and 

n  +

for all closed F and open G, subsets of M1(S). The functional I(&) is lowersemicontinuous in the weak topology; it is called ``rate functional'' or ``level-2 entropy function.'' One says that the process satisfies a level-2 Large Deviation Principle (LDP) with rate functional I( } ). The level-1 LDP for (V n ) n0 deals with means (1n)  n&1 k=0 V k . The analogous to (1.2) and (1.3) have F and G as subsets of S. In a recent work, Carmona et al. (1) proves the existence of a level-2 LDP for a class of Markov processes (V n ) n0 given by V n =(X n , ! n , ! n+1 ),

n0

(1.4)

where ! 1 , ! 2 ,... are i.i.d. random variables with common distribution ' and X n =T n(X 0 )

(1.5)

T n is the group of the iterates of a bijective uniquely ergodic transformation on the unit circle, preserving the Lebesgue measure *, X 0 being the random variable describing the position on the circle and distributed according to *. The processes (X n ) n0 and (! n ) n0 are independent. In the above mentioned article, the final goal was to investigate large deviation properties of the autocorrelation of order k of the process Z n = ,(X n )+! n where , is a continuous real function on the circle. We refer the reader to A. Lopes and S. Lopes (8) for example where such kind of model appears and for general results. By assuming k=1 (in order to simplify the arguments), the method employed in ref. 1 consists, as a first step (and the more difficult one), in obtaining a LDP for the empirical means of (1.4). The second step is to use

Large Deviations for Expanding Transformations

1313

the contraction principle to obtain a LDP for the autocorrelations of order 1 Mn =

1 n&1 : Z j Z j+1 , n j=0

n1

by noticing that Z n Z n+1 is a continuous functional of V n . At level 1 analysis we assume that the measure ' has compact support. Under the above assumptions, the Markov process in (1.4) is ergodic in the sense of having a unique stationary distribution, which is *_'_'. In this case, Carmona et al. (1) established a level 2 LDP for (V n ) n0 in the probability space (S N, _(C), P xyz ) where S=M_R_R is the phase space of the process, _(C) is the _-field of the cylinder sets of S N, and P xyz ( } ) is the distribution of the process when the initial distribution is _ (x, y, z)( } ), (x, y, z) # S. The upper and lower bounds in (1.2) and (1.3), respectively, were obtained by employing Donsker and Varadhan's (3) approach. For Markov processes having a unique stationary distribution, Donsker and Varadhan's method consists, roughly speaking, in proving that I(&)=& inf ,#W

|

ln S

6, d&, ,

& # M1(S)

(1.6)

is the level 2 entropy function of the process. In (1.6) 6,(v)=

|

,(u) 6(v, du)

(1.7)

S

6(v, du) is the transition function given by 6((x, y, z), d(x 1 , y 1 , z 1 ))=$ T (x)(dx 1 ) $ z (dy 1 ) '(dz 1 )

(1.8)

and W=[,: S  R : , is continuous, _a, b such that 0